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%I #10 Apr 11 2022 21:55:14
%S 1,2,4,3,5,9,6,10,7,11,8,12,16,13,17,14,18,15,20,25,19,26,21,27,22,36,
%T 23,30,24,28,37,29,38,31,39,32,40,33,49,34,41,35,42,50,43,51,44,64,45,
%U 65,46,66,47,55,48,56,68,53,72,57,81,52,82,54,83,58,84,59
%N a(1) = 1; for n > 1, a(n) is the smallest positive integer not occurring earlier such that the intersection of the periodic parts of the continued fractions for square roots of a(n) and a(n-1) is the empty set.
%C Conjecture: This is a permutation of the positive integers.
%C The conjecture is true: we can always extend the sequence with a square, so eventuality every square will appear; also, after a square, we can always extend the sequence with the least number not yet in the sequence. - _Rémy Sigrist_, Mar 12 2022
%C The periodic part of the continued fraction for the square root of a square is the empty set.
%e n a(n) Periodic part of continued fraction for square root of a(n)
%e -- ---- -----------------------------------------------------------
%e 1 1 {}
%e 2 2 {2}
%e 3 4 {}
%e 4 3 {1,2}
%e 5 5 {4}
%e 6 9 {}
%e 7 6 {2, 4}
%e 8 10 {6}
%e 9 7 {1, 1, 1, 4}
%e 10 11 {3, 6}
%e 11 8 {1, 4}
%t pcf[m_]:=If[IntegerQ[Sqrt@m],{},Last@ContinuedFraction@Sqrt@m];
%t a[1]=1;a[n_]:=a[n]=(k=2;While[MemberQ[Array[a,n-1],k]||Intersection[pcf@a[n-1],pcf@k]!={},k++];k);Array[a,100]
%Y Cf. A121339, A349637.
%K nonn
%O 1,2
%A _Giorgos Kalogeropoulos_, Feb 16 2022
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